The homogeneous approximation property and localized Gabor frames

Abstract

The homogeneous approximation property (HAP) has been introduced in order to describe the locality of Gabor expansions in the Hilbert space \(L^{2}({\mathbb {R}}^{d})\). In this manuscript the HAP is established for families of modulation spaces. Instead of the more recent theory of localized frames (Grochenig in J Fourier Anal Appl 10(2):105–132, 2004) which relies on Wiener pairs of Banach algebras of matrices, our approach is based on the constructive principles established in Feichtinger and Grochenig (J Funct Anal 86:307–340, 1989, Monatsh Math 108:129–148, 1989), Grochenig (Monatsh Math 112:1–41, 1991), using the fact that generalized modulation spaces are coorbit spaces with respect to the Schrodinger representation of the Heisenberg group (cf. Feichtinger and Grochenig in Wavelets—a tutorial in theory and applications, Academic Press, Boston, pp 359–397, 1992). For the (non-canonical) dual frames obtained constructively in this way the HAP property is verified.